Mixed Nash Equilibrium Research Articles

A mixed Nash equilibrium solution for visibility-based pursuit-evasion game with multiple obstacles

A mixed Nash equilibrium solution for visibility-based pursuit-evasion game with multiple obstacles

Coevolutionary Algorithm for Building Robust Decision Trees under Minimax Regret

In recent years, there has been growing interest in developing robust machine learning (ML) models that can withstand adversarial attacks, including one of the most widely adopted, efficient, and interpretable ML algorithms—decision trees (DTs). This paper proposes a novel coevolutionary algorithm (CoEvoRDT) designed to create robust DTs capable of handling noisy high-dimensional data in adversarial contexts. Motivated by the limitations of traditional DT algorithms, we leverage adaptive coevolution to allow DTs to evolve and learn from interactions with perturbed input data. CoEvoRDT alternately evolves competing populations of DTs and perturbed features, enabling construction of DTs with desired properties. CoEvoRDT is easily adaptable to various target metrics, allowing the use of tailored robustness criteria such as minimax regret. Furthermore, CoEvoRDT has potential to improve the results of other state-of-the-art methods by incorporating their outcomes (DTs they produce) into the initial population and optimize them in the process of coevolution. Inspired by the game theory, CoEvoRDT utilizes mixed Nash equilibrium to enhance convergence. The method is tested on 20 popular datasets and shows superior performance compared to 4 state-of-the-art algorithms. It outperformed all competing methods on 13 datasets with adversarial accuracy metrics, and on all 20 considered datasets with minimax regret. Strong experimental results and flexibility in choosing the error measure make CoEvoRDT a promising approach for constructing robust DTs in real-world applications.

Mixed nash equilibrium and its application in two-player zero-Sum games

The Colonel Blotto game represents a classic case in game theory, embodying the essence of two-player zero-sum games. A vital game theory theorem has demonstrated a mixed Nash equilibrium in all zero-sum games. In this paper, the author will start with this classic game to expound the principles of mixed Nash equilibrium. Furthermore, the paper will explore the practical applications of mixed Nash equilibrium in the context of two-player zero-sum games, illustrating its significance in addressing real-world issues and decision-making. By delving into the Colonel Blotto game and its associated theories, the paper will reveal the importance and broad applicability of mixed Nash equilibrium, showcasing its role in various real-world applications, including competitive strategies, resource allocation, risk management, and more. This research will assist readers in gaining a better understanding of the concept of mixed Nash equilibrium and how to apply it to the resolution of practical problems.

Cooperation in two-player social dilemmas with other-regarding orientations

We present a unified description of the influence of other-regarding orientations (ORO) on the evolution of cooperation and social tensions in all basic classes of two-person social dilemmas. We provide analytical conditions for which two types of ORO, the minimum payoff orientation and the total payoff orientation lead to full cooperation between players. It turns out that in many popular social dilemmas both types of ORO are necessary to promote cooperation, and the minimum payoff orientation may have advantage over the total payoff orientation. Moreover, ORO can be more effective in fostering cooperation in more challenging social dilemmas. For the dilemma games for which the initial and modified through ORO payoff matrices have mixed Nash equilibria we prove that ORO fosters cooperation. Once a real life situation is identified as a particular type of social dilemma in which the pro-social behavior is determined by material payoffs, our results determine the choice of the type and of the intensity of ORO necessary to foster cooperation and to erase social tensions between players.

A methodology for solving bimatrix games under 2-tuple linguistic environment

In a game problem, when the situation is formulated as the gain of one equals the loss of other, the game is called zero-sum. But actually, one player's win is not always the other's loss. Such a game is referred to as non-zero sum or bimatrix game. The work provided in this research is primarily elaborated on linguistic bimatrix games under 2-tuple linguistic environment. This class of bimatrix games consists of payoffs having entries as linguistic 2-tuples, that are based on a finite subscript-symmetric additive linguistic set defined in the domain , where g is a positive integer. In this direction, a novel linguistic equilibrium concept is defined to solve such games, and a methodology based on a nonlinear linguistic programme is proposed to compute mixed Nash equilibrium. The proposed methodology is sufficient to evaluate the Nash equilibrium and the maximum expected payoffs corresponding to each player simultaneously in a single programme. The necessary and sufficient conditions for the existence of Nash equilibrium and a computation procedure are also provided. Finally, some hypothetical working examples are given to show the practical applicability of the defined methodology. The proposed methodology is also applied to solve two-player constant sum linguistic games.

Public goods games in directed networks

Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to prove, finding the equilibrium is PPAD-complete. Finally, when the treewidth of the directed network is appropriately bounded, we prove that polynomial-time algorithms are possible.

STSIR: An individual-group game-based model for disclosing virus spread in Social Internet of Things

Social Internet of Things (SIoT) with deep integration of Internet of Things and social networks has become a target of a large number of hackers who attempt to spread viruses for breaching data confidentiality and service reliability. Therefore, exposing the law of virus spread with social characteristics and addressing historical dependence of infection and recovery rates in an SIoT are urgent problems to be solved at present. To this end, we propose a novel virus spread model (STSIR) based on an epidemic theory's analysis framework and individual-group game theory, which more reasonably describes viruses spread among devices considering people behavior. Aiming at the characteristics of SIoTs including limited social distance and dynamic number variation of people and devices, we adopt and improve the traditional epidemic model SIR to reveal the form of viruses continuously spreading to neighbor nodes. We then introduce an individual-group game to establish the attack and defense model between infected SIoT nodes and susceptible SIoT nodes, in order to not only obtain the mixed Nash equilibrium solution by using a payoff matrix but also solve the dependence of the infection and recovery rates on historical experience. Further, we establish differential equations to represent the model STSIR, which are the basis of proving the existence of model equilibrium points and analyzing stability mathematically. Finally, the effectiveness of the model STSIR in curbing virus spread is verified by simulating two equilibrium points. Under the same conditions in an SIoT, the model STSIR reduce viruses by ∼45% more than the model SIS, and saves stabilization time cost by ∼66.7% compared with the model SIR, which proves that the model STSIR is obviously more effective.

Solving Two-Person Zero-Sum Stochastic Games With Incomplete Information Using Learning Automata With Artificial Barriers.

Learning automata (LA) with artificially absorbing barriers was a completely new horizon of research in the 1980s (Oommen, 1986). These new machines yielded properties that were previously unknown. More recently, absorbing barriers have been introduced in continuous estimator algorithms so that the proofs could follow a martingale property, as opposed to monotonicity (Zhang et al., 2014), (Zhang et al., 2015). However, the applications of LA with artificial barriers are almost nonexistent. In that regard, this article is pioneering in that it provides effective and accurate solutions to an extremely complex application domain, namely that of solving two-person zero-sum stochastic games that are provided with incomplete information. LA have been previously used (Sastry et al., 1994) to design algorithms capable of converging to the game's Nash equilibrium under limited information. Those algorithms have focused on the case where the saddle point of the game exists in a pure strategy. However, the majority of the LA algorithms used for games are absorbing in the probability simplex space, and thus, they converge to an exclusive choice of a single action. These LA are thus unable to converge to other mixed Nash equilibria when the game possesses no saddle point for a pure strategy. The pioneering contribution of this article is that we propose an LA solution that is able to converge to an optimal mixed Nash equilibrium even though there may be no saddle point when a pure strategy is invoked. The scheme, being of the linear reward-inaction ( LR-I ) paradigm, is in and of itself, absorbing. However, by incorporating artificial barriers, we prevent it from being "stuck" or getting absorbed in pure strategies. Unlike the linear reward- ϵ penalty ( LR-ϵP ) scheme proposed by Lakshmivarahan and Narendra almost four decades ago, our new scheme achieves the same goal with much less parameter tuning and in a more elegant manner. This article includes the nontrial proofs of the theoretical results characterizing our scheme and also contains experimental verification that confirms our theoretical findings.

Nash mixed detection strategy of multi-type network attack based on zero-sum stochastic game

The defense strategy of the Internet of Things (IoT) under hacker attack is a crucial issue in the security field of industry. When IoT plants are attacked by hackers, it is impossible to understand the type of attack employed by the hackers. At this point, we need a detection mechanism that can cope with the known multiple types of network attacks. In this paper, we focus on the issue of IoT security defense and construct a multi-detector detection system capable of detecting multiple types of network attacks based on a zero-sum stochastic game framework. The system can choose between multiple detectors according to a mixed Nash equilibrium strategy that takes into account both the detection cost and the detection effect. Finally, we present a simulation comparison between the multi-detector system and the single detector system to verify that the mixed strategy leads to higher detection effectiveness and lower control performance loss.

Adaptive learning with artificial barriers yielding Nash equilibria in general games

Abstract Artificial barriers in Learning Automata (LA) is a powerful and yet under-explored concept although it was first proposed in the 1980s. Introducing artificial non-absorbing barriers makes the LA schemes resilient to being trapped in absorbing barriers, a phenomenon which is often referred to as lock in probability leading to an exclusive choice of one action after convergence. Within the field of LA and reinforcement learning in general, there is a sacristy of theoretical works and applications of schemes with artificial barriers. In this paper, we devise a LA with artificial barriers for solving a general form of stochastic bimatrix game. Classical LA systems possess properties of absorbing barriers and they are a powerful tool in game theory and were shown to converge to game’s of Nash equilibrium under limited information. However, the stream of works in LA for solving game theoretical problems can merely solve the case where the Saddle Point of the game exists in a pure strategy and fail to reach mixed Nash equilibrium when no Saddle Point exists for a pure strategy. Furthermore, we provide experimental results that are in line with our theoretical findings.

Coherence of probabilistic constraints on Nash equilibria

In this work, we first deal with the modeling of game situations that reach one of possibly many Nash equilibria. Before an instance of such a game starts, an external observer does not know, a priori, what is the exact profile of actions -- constituting an equilibrium -- that will occur; thus, he assigns subjective probabilities to players' actions. Such scenario is formalized as an observable game, which is a newly introduced structure for that purpose. Then, we study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, called the PCE-Coherence problem. We show several results concerning algorithms and complexity for PCE-Coherence when pure Nash equilibria and specific classes of games, called GNP-classes, are considered. In this context, we also study the computation of maximal and minimal probabilistic constraints on actions that preserves coherence. Finally, we study these problems when mixed Nash equilibria are allowed in GNP-classes of 2-player games.

Decentralized payment clearing using blockchain and optimal bidding

In this paper, we construct a decentralized clearing mechanism which endogenously and automatically provides a claims resolution procedure. This mechanism can be used to clear a network of obligations through blockchain. In particular, we investigate default contagion in a network of smart contracts cleared through blockchain. In so doing, we provide an algorithm which constructs the blockchain so as to guarantee the payments can be verified and the miners earn a fee. We, additionally, consider the special case in which the blocks have unbounded capacity to provide a simple equilibrium clearing condition for the terminal net worths; existence and uniqueness are proven for this system. Finally, we consider the optimal bidding strategies for each firm in the network so that all firms are utility maximizers with respect to their terminal wealths. We show that a pure strategy Nash equilibrium need not exist, and then look for mixed Nash equilibrium bidding strategies. The implications of these strategies, and more broadly blockchain, on systemic risk are considered.

Fictitious Play in Zero-Sum Stochastic Games

We present fictitious play dynamics for stochastic games and analyze its convergence properties in zero-sum stochastic games. Our dynamics involves players forming beliefs on opponent strategy and their own continuation payoff (Q-function), and playing a greedy best response using estimated continuation payoffs. Players update their beliefs from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update of the beliefs on strategies. We show both in the model-based and model-free cases (without knowledge of player payoff functions and state transition probabilities), the beliefs on strategies converge to a stationary mixed Nash equilibrium of the zero-sum stochastic game.

Rumor Transmission in Online Social Networks Under Nash Equilibrium of a Psychological Decision Game.

This paper investigates rumor transmission over online social networks, such as those via Facebook or Twitter, where users liberally generate visible content to their followers, and the attractiveness of rumors varies over time and gives rise to opposition such as counter-rumors. All users in social media platforms are modeled as nodes in one of five compartments of a directed random graph: susceptible, hesitating, infected, mitigated, and recovered (SHIMR). The system is expressed with edge-based formulation and the transition dynamics are derived as a system of ordinary differential equations. We further allow individuals to decide whether to share, or disregard, or debunk the rumor so as to balance the potential gain and loss. This decision process is formulated as a game, and the condition to achieve mixed Nash equilibrium is derived. The system dynamics under equilibrium are solved and verified based on simulation results. A series of parametric analyses are conducted to investigate the factors that affect the transmission process. Insights are drawn from these results to help social media platforms design proper control strategies that can enhance the robustness of the online community against rumors.

A convergence analysis of the price of anarchy in atomic congestion games

We analyze the convergence of the price of anarchy (PoA) of Nash equilibria in atomic congestion games with growing total demand T. When the cost functions are polynomials of the same degree, we obtain explicit rates for a rapid convergence of the PoAs of pure and mixed Nash equilibria to 1 in terms of 1/T and d_{max}/T, where d_{max} is the maximum demand controlled by an individual. Similar convergence results carry over to the random inefficiency of the random flow induced by an arbitrary mixed Nash equilibrium. For arbitrary polynomial cost functions, we derive a related convergence rate for the PoA of pure Nash equilibria (if they exist) when the demands fulfill certain regularity conditions and d_{max} is bounded as Trightarrow infty . In this general case, also the PoA of mixed Nash equilibria converges to 1 as Trightarrow infty when d_{max} is bounded. Our results constitute the first convergence analysis for the PoA in atomic congestion games and show that selfish behavior is well justified when the total demand is large.

Learning Automata with Artificial Reflecting Barriers in Games with Limited Information

This paper deals with the problem of solving stochastic games (which have numerous business and economic applications), using the interesting tools of Learning Automata (LA), the precursors to Reinforcement Learning (RL). Classical LA systems that possess properties of absorbing barriers, have been used as powerful tools in game theory to devise solutions that converge to the game's Nash equilibrium under limited information(Sastry, Phansalkar, and Thathachar 1994). Games with limited information are intrinsically hard because the player does not know the actions chosen of other players, neither their outcomes. The player might not be even aware of the fact that he/she is playing against an opponent.
 With the state-of-the-art, the numerous works in LA applicable for solving game theoretical problems, can merely solve the case where the game possesses a Saddle Point in a pure strategy.
 They are unable to reach mixed Nash equilibria when a Saddle Point is non-existent in pure strategies. Additionally, within the field of LA and RL in general, the theoretical and applied schemes of LA with artificial barriers are scarce, even though incorporating artificial barriers in LA has served as a powerful and yet under-explored concept, since its inception in the 1980’s. More recently, the phenomenon of introducing artificial non-absorbing barriers was pioneered, and this renders the LA schemes to be resilient to being trapped in absorbing barriers. In this paper, we devise a LA with artificial barriers for solving a general form of stochastic bimatrix games. The problem’s complexity has been augmented with the scenario that we consider games in which there is no Saddle Point. By resorting to the above-mentioned powerful concept of artificial reflecting barriers, we propose a LA that converges to an optimal mixed Nash equilibrium even though there may be no Saddle Point when a pure strategy is invoked.

A noncooperative game‐theoretic approach applied to the service selection in the vehicular cloud

SummaryRecent advancements in the automotive area have aided in the development of smart vehicles with high processing, computing, and storage capabilities. These resources are leveraged to cope with the demands of consumer vehicles such as data analysis, optimization, and decision‐making. The first and foremost step in establishing this paradigm of computing so‐called vehicular cloud is the service selection. Nonetheless, vehicles' high mobility and the network's rapidly changing topology have addressed new challenges such as instability of resources, making their management complicated. Therefore, the vehicular cloud requires robust and efficient resource management solutions to be developed. In this article, we present a game theory‐based framework for vehicular resource selection where provider vehicles and consumer vehicles play a game against each other until reach equilibrium in order to enable consumer vehicles to select the most suitable provider vehicles that meet their requirements in terms of service quality and cost. Furthermore, a thorough study was carried out in order to study the interaction between players and predict their future behaviors. As this interaction will result in a pure and mixed equilibrium state, the convergence of the proposed theoretic model for both equilibrium states is rigorously demonstrated under certain assumptions. Effective search algorithms based on the best responses are designed to compute the pure and mixed Nash equilibrium. The suggested approach's verification is evaluated based on the satisfaction of the players and has shown better performance comparing previous algorithms.

Mixed Nash equilibria for continuous games and reverse mathematics

The program reverse mathematics seeks to identify the minimal axioms needed to prove theorems of ordinary mathematics. In this paper, we develop the reverse mathematics of measure theory and apply these results to the reverse mathematics study of game theory. We address the technical issue of how to develop product measures in reverse mathematics, including formalizing Fubini’s thoerem. We show that weak compactness of probability measures on a compact space is equivalent to arithmetical comprehension over RCA0. The forward direction is again slightly technical. As an application of these results, we prove in ACA0 that any continuous game has a mixed Nash equilibrium.

Solving possibilistic games with incomplete information

Bayesian games offer a suitable framework for games where the utility degrees are additive in essence. This approach does nevertheless not apply to ordinal games, where the utility degrees do not capture more than a ranking, nor to situations of decision under qualitative uncertainty. The present paper proposes a representation framework for ordinal games under possibilistic incomplete information and extends the fundamental notions of pure and mixed Nash equilibrium to this framework. We show that deciding whether a pure Nash equilibrium exists is a difficult task (NP-hard) and propose a Mixed Integer Linear Programming (MILP) encoding of the problem; as to the problem of computing a possibilistic mixed equilibrium, we show that it can be solved in polynomial time. An experimental study based on the GAMUT game generator confirms the feasibility of the approach.

General features of Nash equilibria in combinations of elementary interactions in symmetric two-person games

Two-person games are used in many multi-agent mathematical models to describe pair interactions. The type (pure or mixed) and the number of Nash equilibria affect fundamentally the macroscopic behavior of these systems. In this paper, the general features of Nash equilibria are investigated systematically within the framework of matrix decomposition for n strategies. This approach distinguishes four types of elementary interactions that each possess fundamentally different characteristics. The possible Nash equilibria are discussed separately for different types of interactions and also for their combinations. A relation is established between the existence of infinitely many mixed Nash equilibria and the zero-eigenvalue eigenvectors of the payoff matrix.